Question 1: Find the perimeter and area of a rectangle whose length and breadth are and
respectively.
Answer:
Dimensions of the rectangle: Length Breadth
Therefore Perimeter
Area
Question 2: A rectangular room floor is in area. If its length is
, find its perimeter.
Answer:
Dimensions of the rectangle: Length Let Breadth
Area of rectangle
Hence Breadth
Question 3: Find the length of a diagonal of a rectangle whose adjacent sides are and
long.
Answer:
Dimensions of the rectangle: Length Let Breadth
Diagonal of a rectangle
Question 4: Find the length of a diagonal of a square of side .
Answer:
Dimension of a square: Side
Diagonal of a square
Question 5: Find the perimeter of a square the sum of the lengths of whose diagonal is .
Answer:
Dimension of a square: Side
Given: Diagonal of the square
We know diagonal of a square
Perimeter of a square
Question 6: The length and breadth of a room are in the ratio . Its area is
. Find its perimeter.
Answer:
Dimensions of the rectangle: Let Length Let Breadth
Given: Area is
Therefore
Hence Length Let Breadth
Therefore Perimeter
Question 7: The diagonal of a square is
. Find the diagonal of a square
whose area is twice the area of
.
Answer:
Given: Diagonal of a square is
If the side of the square
… … … … … (i)
Let the side of the second square
Given:
Diagonal … … … … … (ii)
Substituting (i) in (ii) we get
Diagonal
Question 8: The perimeter of a square is . Find its diagonal.
Answer:
Perimeter
Hence Diagonal
Question 9: Find the area of a square that can be inscribed in a circle of radius .
Answer:
Radius
Hence the length of the side
Therefore Area
Question 10: Find the perimeter of a square the sum of the lengths of whose diagonals is .
Answer:
Let the side be
Given:
Therefore Perimeter
Question 11: The diagonal of a square is . Find its area.
Answer:
Let the side of the square
Given:
Therefore Area
Question 12: Find the area and perimeter of a square plot of land the length of whose diagonal is .
Answer:
Given: Diagonal
Let the side of the square
Therefore
Therefore Area
Hence the Perimeter
Question 13: Find the ratio of the area of a square to that of the square drawn on its diagonal.
Answer:
Let the side of square
Therefore diagonal
Hence the ratio
Question 14: The diagonal of square is
. Find the diagonal of square
whose area is half of the area of
.
Answer:
Let the side of square
Therefore
Let the side of square
Therefore
Diagonal of square
Question 15: The perimeter of a square is . The area of a rectangle is
less than the area of the given square. If the length of the rectangle is
, find its breadth.
Answer:
Let the side of the square
Therefore
Let the breadth of the rectangle
Therefore
Question 16: The perimeter of one square is and that of another is
. Find the perimeter and the diagonal of a square whose area is equal to the sum of the areas of these two squares.
Answer:
Let the side of square 1
Therefore 4
Hence the area of square 1
Let the side of square 2
Therefore 4
Hence area of square 2
Therefore area of square 3
Hence the side of square 3
Therefore the perimeter of square 3
Diagonal of square 3
Question 17: The perimeter of a rectangular card board is . If its breadth is
, find the length and area of the card board.
Answer:
Dimensions of the rectangle: Length Breadth
Therefore
Hence area
Question 18: If the sides of two squares are in the ratio , prove that their areas are in the ratio
.
Answer:
Side of square 1
Therefore Area of square 1
Side of square 2
Therefore Area of square 2
Therefore ratio of areas
Question 19: In exchange for a square plot one of whose sides is , a man wants to buy a rectangular plot
long and of the same area as of the square plot. Find the width of the rectangular plot.
Answer:
Side of square plot
Length of rectangular plot
Let breadth of rectangular plot
Therefore
Question 20: A rectangular lawn has two roads each with
wide running in the middle of it, one parallel to the length and other parallel to the breadth. Find the cost of graveling them at
paisa per square meter.
Answer:
Area of graveled road
Therefore cost of graveling
Question 21: The area of a square plot is hectare. Find the diagonal of the square.
Answer:
Let the side of square plot
Therefore hectare
We know 1 hectare
Therefore
Hence diagonal
Question 22: A lawn is in the form of a rectangle having its sides in the ratio . The area of the lawn is
. Find the cost of fencing it at the rate of
per meter.
Answer:
Let length and breadth
Therefore
Therefore length and breadth
Therefore perimeter
Therefore cost of fencing
Question 23: The area of a square park is . Find the cost of fencing it at the rate of
per meter.
Answer:
Area of square
Let side of the square
Therefore Perimeter
Therefore cost of fencing
Question 24: The area of the base of a rectangular tank is and its sides are in the ratio
. Find the cost of planting flowers round it at the rate of
per meter.
Answer:
Let length and breadth
Therefore
Therefore length and breadth
Therefore perimeter
Therefore cost of planting flowers
Question 25: A rectangular field meter long has got an area of
, what will be the cost of fencing that field on all the four sides, if
meter of fencing costs
paisa?
Answer:
Length
Area
Therefore breadth
Perimeter
Therefore cost of fencing
Question 26: A rectangular grassy plot is . It has gravel path
wide all around it on the inside. Find the area of the path and the cost of constructing it at the rate of
per sq. meter.
Answer:
Dimensions of park: Length , Breadth
Inner Dimensions of park: Length , Breadth
Area of path
Therefore cost of construction
Question 27: There is a square field whose side is . A flowerbed is prepared in its center, leaving a gravel path of uniform width all around the flower bed. The total cost of laying the flower bed and graveling the path at
and
per square meter respectively is
. Find the width of the gravel path.
Answer:
Let the side of the garden
Therefore the area of the garden
Area of the path
Therefore
Hence the path
Question 28: How many tiles each will be required to pave the footpath
wide carried round the outside of a grassy plot
by
?
Answer:
Are of footpath
No of times required
Question 29: A room is long,
broad and
high. It has two doors
and
windows each
. How much will it cost to whitewash the walls of the room at the rate of
per square meter?
Answer:
Dimensions of the Room: Length (l) , Breadth
and Height
Dimensions of the Doors: Breadth and Height
Dimensions of the Window: Breadth and Height
Area of walls = Lateral Area
Area of Doors and Windows
Area to be painted
Cost of painting
Question 30: A carpet is laid on the floor of a room . There is a border of constant width around the carpet. If the area of the border is
, find its width.
Answer:
Dimensions of the Room: Length (l) , Breadth
Area of Border
Therefore
or
(this is not possible as
cannot be negative)
Question 31: A rectangular courtyard, long and
broad, is to be paved exactly -with square tiles, all of the same size. Find the largest size of such a tile and the number of tiles required to pave it.
Answer:
Dimension of courtyard: Length
Breadth
The largest common divisor of both the numbers is
Therefore the dimension of the largest square tile is
Hence the number of tiles required
Question 32: The cost of fencing a square field at paisa per meter is
. Find the cost of reaping the field, it the rate of
\ paisa per
.
Answer:
Perimeter
Therefore side
Therefore Area
Therefore cost of reaping the field
Question 33: A room long and
broad is carpeted with a carpet, leaving an uncovered margin
all around the room. If the breadth of the carpet is
, find its cost at
per meter.
Answer:
Dimensions of the Room: Length (l) , Breadth
Dimensions of the Carpet: Length (l) , Breadth
Area
Let the length of the carpet needed
Therefore
Therefore cost
Question 34: The cost of carpeting a room at Per square meter is
. The cost of whitewashing the walls at
paisa per sq. meter is
. The room is
wide. Find its height.
Answer:
Cost of paining area of walls
Question 35: A room long and
wide is surrounded by a verandah. Find the width of the verandah if it occupies
square meters.
Answer:
Let be the width of the varanda
Therefore
or
(not possible)
Therefore the width of the verandah
Question 36: Square carpet is spread in the center of a room square leaving a small margin of equal width all around. The total cost of carpeting at
and decorating the margin at
is
. Find the width of the margin.
Answer:
Dimension of square room
Let the width of the margin
Therefore area of carpet
Area of margin
(not possible)
Hence the width of the margin
Question 37: If the length and breadth of a room are increased by the area is increased By
. If the length is increased by
and breadth is decreased by
, the area is decreased by
. Find the perimeter of the room.
Answer:
… … … … … i)
… … … … … ii)
From i)
… … … … … iii)
Therefore Perimeter
from ii)
… … … … … iv)
Solving (iii) and (iv) we get
and
.
Question 38: A rectangle has twice the area of a square. The length of the rectangle is greater and the width is
greater than the side of the square. Find the perimeter of the square.
Answer:
Let the dimension of rectangle be and
. And that of the square be
.
Therefore and
Therefore Perimeter of square
Question 39: If the perimeter of a rectangular plot is and the length of its diagonal is
, find its area.
Answer:
Let the dimension of rectangle be and
.
Therefore … … … … … i)
… … … … … ii)
Therefore
Substituting in (ii)
Hence the area is
Question 40: The length of a rectangular garden is more than its breadth. The numerical value of its area is equal to
times the numerical value of its perimeter. Find the dimensions of its garden.
Answer:
Area
Perimeter
Therefore
Therefore or
(not possible)
Hence . Therefore dimensions are
and
Question 41: A wire when bent in the form of an equilateral triangle encloses an area of Find the area enclosed by the same wire when bent to form: (i) a square (ii) a rectangle whose length is
more than its width.
Answer:
Therefore Perimeter
i) When bent in a square
Side
Therefore area
ii) Perimeter of rectangle
If Breadth
Then
Therefore dimensions are and
Hence Area